Signals are useful for a variety of applications. Some applications involve simultaneously generating and using multiple signals. Sometimes signals propagating through a medium can (at least partially) interfere with each other. For example, electromagnetic signals exhibit this behavior. This interference can sometimes be undesirable. For example, when the signals serve as information carriers, the interference can lead to loss of information contained in one or more of the signals.
One example application involves multiple-input, multiple-output (“MIMO”) detection systems. In particular, MIMO radar is an emerging technology that is attracting the attention of researchers and practitioners alike. Among recent efforts in this area is the design of a multi-waveform radar system for urban warfare. The popularity of radar is at least in part attributable to improved capabilities of multi-beam radar, as compared with standard phased-array and single-input multiple-output (SIMO) radar.
FIGS. 1A and 1B show an exemplary MIMO radar system used to detect airborne vehicles or objects, although a similar radar system can be designed for use in urban warfare. This exemplary MIMO radar system includes a transmitter subsystem 141 (FIG. 1A) and a receiver subsystem 142 (FIG. 1B). The transmitter subsystem 141 includes a code sequence store 111, which can store orthogonal code sequences or any other code sequences with desired properties, e.g., cross-correlation properties. The transmitter subsection also includes multiple digital modulators 115a-115c, each of which modulates a different code from the code sequence store 111. The digital modulators 115a-115c can use any form of digital modulation such as Phase Shift Keying (PSK). The transmitter subsystem 141 also includes an oscillator 113 for generating an RF signal and multiple frequency mixers or up converters 117a-117c. The frequency mixers 117a-117b modulate the RF signal from the oscillator 113 with modulated codes from the digital modulators 115a-115c to generate multiple modulated RF signals. The transmitter subsystem 141 includes multiple transmitters 119a-119c that radiate the modulated RF signals as electromagnetic signals 131.
If there is an object 150 (e.g., an airborne vehicle) in the path of the signals 131, a portion of each signal 132 is scattered off object 150 and nearby objects 151, 152 (e.g., airborne vehicles) towards multiple receivers 129a-129c of the receiver subsystem 142 (FIG. 1B). The receiver subsystem includes multiple mixers or down converters 127a-127c that down convert the RF signals 132 received by the multiple receivers 129a-129c to baseband frequency using the RF signal generated by the oscillator 123. The receiver subsystem 142 also includes matched filters 125a-125c that filter the down-converted RF signals 132 based on the codes in the code sequence store 111 to detect the codes in the down-converted RF signals 132. The filtered signals are then summed by a summer 121 and passed to a detector 130, such as a peak search and threshold detector. If the detector 130 detects the object 150, it can generate a signal indicating that the object 150 has been detected and determine the location of the object 150.
A MIMO configuration sometimes offers advantages over alternative configurations. For example, a MIMO detection system (radar, sonar, etc.) makes it possible to use adaptive localization and detection techniques. In addition, the probing signal vector transmitted by a MIMO radar system can be designed to approximate a desired transmit beam pattern and to minimize the cross-correlation of the signals bounced from various targets of interest. MIMO radar has potential for fading mitigation, resolution enhancement, and interference and jamming suppression.
Fully exploiting these potentials can result in significantly improved target detection, parameter estimation, and target tracking and recognition performance. To realize these advantages, the information contained in individual components of the echo should be unambiguously retrieved at the receiver. At the same time, to maintain an acceptable receiver signal-to-noise ratio, a waveform set must make optimal use of the available bandwidth. To accommodate these two constraints, one must identify a waveform set that is sufficiently large and whose individual members interfere with each other as little as possible. Techniques exist for developing appropriate waveform sets, however, these techniques are only useful to develop a limited number of waveforms for particular applications.
Another example application that involves simultaneously generating and using multiple signals is wireless communications. Multicarrier techniques are used in wireless communications systems to improve the bandwidth efficiency and reduce the inter-symbol interference of the communications system. Several techniques divide the available portion of the usable electromagnetic spectrum among many channels, and/or transmit data with multiple carriers. Multiple access techniques can be used to increase the number of users that may access the wireless services provided by the system at any given time. Multicarrier and multiple access techniques can also be combined to serve many users simultaneously and provide them with the bandwidth efficiency and wireless services they desire. Examples of multiple access techniques include: Orthogonal Frequency Division Multiplexing (OFDM), Frequency Division Multiple Access (FDMA), Time Division Multiple Access (TDMA), and Code Division Multiple Access (CDMA) and its many variations. Examples of combined multicarrier/multiple access techniques include Orthogonal Frequency Division Multiple Access (OFDMA), Multicarrier Code Division Multiple Access (MC-CDMA), and Multicarrier Direct Sequence CDMA (Multicarrier DS-CDMA).
FIG. 2A is a circuit block diagram of an exemplary MC-CDMA system. The MC-CDMA system includes a transmitter subsystem 210 and a receiver subsystem 220. The transmitter subsystem 210 includes a digital modulator 218 that modulates a given digital input signal to obtain a symbol and a “Copy” unit 217 that copies the symbol to multiple subcarriers. The transmitter subsystem 210 further includes a frequency spread unit 215 that applies spread-spectrum code sequences C0, C1, . . . , CN to each of the subcarriers. These code sequences can be designed to have specified auto- and cross-correlation properties. The transmitter subsystem 210 also includes an Inverse Fast Fourier Transform (FFT) unit that computes the Inverse FFT of the coded subcarriers and a guard insertion unit 212 that inserts guard intervals. The transmitter subsystem 210 lastly includes an antenna 211 that transmits the resulting signal.
The receiver subsystem 220 includes an antenna 221, a guard interval deletion unit 222, an FFT unit 223, a frequency reverse spread unit 225, a synthesis unit 226, and a demodulator 228. When the antenna 221 receives a signal, the guard interval deletion unit 222 removes a guard interval from the signal and the FFT unit 223 computes the FFT of the resulting signal. The frequency reverse spread unit 225 then applies codes to the signal transformed by the FFT unit 223 and the synthesis unit 226 synthesizes the result to obtain symbols. The demodulator 228 then demodulates those symbols to obtain digital data.
FIG. 2B is a circuit block diagram of an exemplary multicarrier DS-CDMA system. The multicarrier-CDMA system also includes a transmitter subsystem 230 and a receiver subsystem 230. The transmitter subsystem 230 includes a serial-to-parallel conversion unit 235, multiple DS-CDMA modulators 233a-233n, respective RF frequency mixers 232a-232n, a summer unit 237, and an antenna 231. The serial-to-parallel conversion unit 235 takes a serial input signal and converts it into multiple parallel signals to be input to respective DS-CDMA modulators 233a-233n. 
The DS-CDMA modulators 233a-233n apply spreading code sequences with desired properties to the parallel signals. Each RF frequency mixer 232a-232n then modulates an associated subcarrier f0, . . . , fn with a respective coded signal. The subcarriers f0, . . . , fn may be designed to be orthogonal to each other to reduce the spacing between subcarriers and to improve utilization efficiency. Then, the summer unit 237 sums the modulated subcarriers and the resulting signal is transmitted from antenna 231.
The receiver subsystem 240 includes an antenna 241, RF frequency mixers 242a-242n, DS-CDMA demodulators 243a-243n, and a parallel-serial conversion unit 245. The RF frequency mixers 242a-242n modulate a signal received from antenna 241 with subcarriers f0, . . . , fn to obtain multiple coded signals. Each DS-CDMA demodulator 243a-243n then demodulates a coded signal from a respective RF frequency mixer 242a-242n using an appropriate code. Finally, the parallel-to-serial conversion unit 245 serializes the demodulated signals.
Each of the techniques and systems described above (and other communications techniques and systems) depend on the auto-correlation and cross-correlation properties of the deployed code sequences or spreading code sequences. For these applications, desirable auto-correlation properties often come at the expense of undesirable cross-correlation properties. Therefore, when designing a wireless multicarrier/multiple access communications system, the auto-correlation and cross-correlation properties need to be optimized.
There are many specific techniques and tools for designing signals or code sequences for use in a variety of applications, including MIMO radar and wireless communications systems. Some of these techniques involve designing finite sequences that meet desired properties, such as desired auto- and cross-correlation properties. Indeed, perfect sequences (i.e., sequences having perfect auto- and cross-correlation properties) have been designed for linear system parameter identification, real-time channel evaluation, direct-sequence spread-spectrum direct access, and frequency-hopped spread-spectrum direct access. These tools, however, are only useful to develop a limited number of signals or waveforms for specific implementations and applications.